Spontaneous and stimulated Raman scattering is the interaction of light with “optical phonons”.

Spontaneous and stimulated Brillouin scattering is the interaction of light with “acoustic phonons” and typically involves counterpropagating waves.
Brillouin and Raman scattering are related to the non-instantaneous part of the third-order nonlinear response.

Two-photon absorption is a process where two photons are simultaneously absorbed, leading to an excitation for which a single photon energy would not be sufficient.
Its strength is related to the imaginary part of the χ(3) tensor, and is generally large for semiconductor media with small bandgap energy.

There are also various other effects which are not directly based on optical nonlinearities, but are nevertheless affecting optical phenomena:

In optical fiber technology, optical nonlinearities are of high interest.
In fibers there is a particularly long interaction length combined with the high intensity resulting from a small mode area.
Therefore, nonlinearities can have strong effects in fibers.
Particularly the effects related to the χ(3) nonlinearity – Kerr effect, Raman scattering, Brillouin scattering – are often important, despite the relatively weak intrinsic nonlinear coefficient of silica: either they act as essential nonlinearities for achieving certain functions (e.g. pulse compression), or they constitute limiting effects in high-power fiber lasers and amplifiers.

Usually, the strength of nonlinear effects is determined by the peak power.
However, there are cases where stronger effects occur for lower peak powers, as explained in a Spotlight article.

Strong nonlinearities also occur at intensities which are high enough to cause ionization in the medium.
This can lead to optical breakdown, possibly even associated with laser-induced damage of the material.
In gases, extremely high optical intensities can be applied, which can lead e.g. to high harmonic generation.

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